Phase transitions in systems with long-range potential

The diagram method is used to study phase transitions in systems with $R\to\infty$ Where $R$ is the range of the attractive potential between particles. If the thermodynamic functions are to be calculated correctly in the neighborhood of a phase transition, it is necessary to allow for diagrams with many vertices and lines. To allow for their contribution, a recursion relation is obtained; it relates diagrams of different orders and structures. The relation is used to estimate the contribution from all the many-vertex diagrams and to obtain a differential equation for $p(\mu,T)$ that is valid as $R\to\infty$ ($p$ is the pressure, $T$ the temperature, and $\mu$ the chemical potential). The solution is investigated for the example of the Ising model. In the two-phase region the $s(H)$ curve does not exhibit the unphysical region with negative susceptibility found in the Curie–Weiss approximation ($s$ is the polarization, $H$ the magnetic field). It follows from the solution that is found that the point $R=\infty$ is an essential singularity, so that the thermodynamic functions cannot be expanded in a Taylor series in powers of $1/R^3$ at points near the phase transition. It is shown that allowing for many-vertex diagrams is equivalent to having an effective interaction between the particles of the “all with all” type that is independent of the mutual separations of the particles.